Notes on Hume's Treatise

by G. J. Mattey

Book I, Part II OF THE IDEAS OF SPACE AND TIME

§ II. Of the infinite divisibility of space and time

A general rule: when ideas are adequate representations of objects, what holds of the ideas holds of the objects, "and this we may in general observe to be the foundation of all human knowledge." Since our ideas are adequate representations of the smallest parts of extension, and the parts can never be inferior to the minima, "What appears impossible and contradictory upon the comparison of these ideas, must be really impossible and contradictory, without any farther excuse or evasion."

What is infinitely divisible contains an infinite number of parts, and reductio can be applied. A finite extension contains infinitely many parts, so no finite extension can be infinitely divisible if this is a contradiction. But the supposition that a finite extension contains infinitely many parts is "absurd." The least ideas Hume can form, when multiplied enough, "at last . . . swells up to a considerable bulk, greater or smaller in proportion as I repeat more or less the same idea." To carry the process on in infinitum wourld require that "the idea of extension must also become infinite." Given that the idea of an infinite number of parts is the same as that of an infinite extension, and a finite extension cannot contain an infinite number of parts, no finite extension is infinitely divisible. (The distinction between proportional and aliquot parts is frivolous.)

Another argument, due to Malizieu, is "very strong and beautiful." The idea is that existence belongs only to unity, and with infinite divisibility, no unities are ever arrived at. If you stopped at a given point, it would be only attributing unity to an aggregate, such as considering twenty men, the whole earth or the whole universe as a unity. "The unity which can exist alone, and whose existence is necessary to that of all number, is of another kind, and must be perfectly indivisible, and incapable of being resolved into any lesser unity."

Application of same argument to time: each of its parts are successive and cannot be co-existent. But if time were infinitely divisible, there would be an infinite number of co-existing parts of time, "which I believe will be allo'd to be an arrant contradiction."

If space is infinitely divisible, so is time, as can be seen from the nature of motion, so if time is not infinitely divisible, neither is space.

A contrary demonstration is not a mere difficulty, as would be claimed by the defenders of infinite divisibility. "A demonstation, if just, admits of no opposite difficulty; and if not just, 'tis a mere sophism, and consequently can never be a difficulty. 'tis irresistible, or has no manner of force." This is not a dialectical situation where arguments pro and contra are balanced against each other.

Mathematicians claim to have equally strong arguments against indivisible points. Before answering them in detail, Hume marshals "a short and decisive reason to prove at once, that 'tis utterly impossible they can have any just foundation."

It is "an establish'd maxim in metaphysics" that nothing we imagine is absolutely impossible. We have an idea of extension, and this idea, "as conceiv'd by the imagination" is not infinitely divisible, as this exceeds our capacities. We produce the ideas, and the idea consequently implies no contradiction. So it is possible that indivisible points exist, and "all the arguments employ'd against the possiblity of mathematical [sic] points are mere scholastick quibbles, and unworthy of our attention."

A further consequence is that all pretended demonstartions of infinite divisiblity are sophistical, because they would have to prove that mathematical points are impossible.

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